Soil-structure interaction of end- frames for high-speed railway bridges

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IN

DEGREE PROJECT CIVIL ENGINEERING AND URBAN MANAGEMENT,

SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2016 ,

Soil-structure interaction of end- frames for high-speed railway bridges

JOHAN ÖSTLUND

KTH ROYAL INSTITUTE OF TECHNOLOGY

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Johan Östlund c

Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering

Division of Structural Engineering and Bridges

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Abstract

In this thesis, the influence of soil-structure interaction (SSI) of end-frame bridges for high-speed railways was studied. Impedance functions, representing the SSI, was calculated and analyzed. The impedance functions were applied to end-frame bridge models which were analyzed for use in HSR.

A new high-speed railway link is currently being planned in Sweden by the Swedish Transport Administration (Trafikverket). Ostlänken is planned to run between the cities of Stockholm and Linköping with a maximum speed limit of 320km/h. As high- speed traffic induces high dynamic impact on bridges, dynamic analysis to ensure safety and passenger comfort is needed according to Eurocode. Thus, there is a demand of dynamically safe bridges that are also cost-effective. One cost-effective bridge is the soil integrated end-frame bridge, however, there are no design advice in Eurocode today on how to take SSI into consideration. The aim of the thesis has therefore been to investigate if the influence of SSI on end-frame bridges for HSR.

This thesis was executed using the frequency domain approach to solve dynamic problems in finite element software. Furthermore, impedance functions have been obtained representing the SSI. Impedance functions take dynamic stiffness and dy- namic damping into consideration where the damping consists of two parts: mate- rial damping and radiation damping due to energy dissipation in the form of elastic waves. To limit the model size, an absorbing region (AR) was used to mitigate waves originating from the source. The accuracy of impedance functions is dependent on several parameters and demands a great computational capacity to reach, mostly governed by the radiation condition. A parameter study of impedance functions was conducted, including parameters such as geometry, modulus of soil and detail lev- els. The impedance functions were then attached to bridge models on which trains modelled as moving point loads were applied. Envelopes of the acceleration and displacements have been presented and analyzed. Shear strain checks were made in order to verify the assumption of linear-elastic material behavior of the embankment.

By using SSI in form of impedance functions attached to bridge models, numerical results show a great reduction of vibrations in models. The study suggests that a large end-frame, either long or high or both, may reduce acceleration as well as displacements. A stiffer embankment material may further reduce vibrations. Shear strain checks confirm that the assumption of linear-elastic soil behavior was true.

Keywords: Dynamics, SSI, Soil-structure interaction, Soil-bridge interaction, HSR,

High-speed railway, Impedance, Receptance, End-frame bridge

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Sammanfattning

I det här exjobbet har påverkan av jord-struktur interaktion (soil-structure interac- tion - SSI) av ändskärmsbroar för höghastighetsbana blivit studerat. Impedansfunk- tioner som representerar SSI har beräknats och analyserats. Impdansfunktionerna har sedan applicerats på bromodeller och analyserats för höghastighetstrafik.

Sveriges första höghastighetsbana håller just nu på att planeras av Trafikverket.

Ostlänken kommer att bli den första delen och är planerad att gå från Stockholm till Linköping med en högsta hastighet av 320 km/h. Då höghastighetstrafik intro- ducerar stor dynamisk påverkan på broar behövs dynamisk analys genomföras enligt Eurocode för att kunna säkerställa broarnas säkerhet och komfortkrav. Därför finns idag ett behov av dynamiskt säkra broar som också är kostnadseffektiva. En typ av kostnadseffektiv bro är den med jord integrerade ändskärmsbron. I dagens Eurocode finns dock inga konstruktionsråd vad gäller jord-struktur interaktion av ändskär- marna. Målet med detta examensarbete har därför varit att undersöka påverkan av SSI och besluta huruvida användandet av ändskärmsbron på höghastighetsbanor är legitimerat, eller om den ska undvikas.

Det här examensarbetet har utgått från att lösa dynamiska problem i frekvensdomä- nen med hjälp av FEM. Impedansfunktioner som representerar jord-struktur inter- aktionen har tagits fram. Impedansfunktioner tar dels hänsyn till dynamisk styvhet och dels dynamisk dämpning. Den dynamiska dämpningen består av två delar;

den första är materialdämpning och den andra är vågdämpning där energi dis- siperar i vågform. För att begränsa FE modellens storlek har en absorbing region tillämpats för att absorbera vågorna vid randen. Impedansfunktionernas konvergens beror på flertalet parametrar och kräver en hög datakapacitet för att fås, mestadels beroende av radiatorvillkoret. En parameterstudie utfördes för att kunna analysera sensitiviteten hos impedansfunktionerna. Vidare applicerades dessa impedansfunk- tioner på skal- och balk-bromodeller på vilka HSLM laster påfördes. Skjuvtöjn- ingskontroller gjordes för att verifiera att antagandet om linjärelastiskt materialbe- teende var korrekt.

Genom att ta hänsyn till SSI i form av impedansfunktioner tyder numeriska resultat på att vibrationer kan reduceras i hög grad. Envelopper visar att en stor ändskärm, antingen lång, hög eller bådadera, kan reducera accelerationer liksom förskjutningar.

En styvare bank kan ytterligare reducera vibrationer.

Sökord: Dynamik, SSI, Jord-struktur interaktion, Jord-bro interaktion, HSR,

Höghastighetsbana, Höghastighetståg, Impedans, Receptans, Ändskärmsbro

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Preface

This master thesis was initiated by the Department of Civil and Architectural En- gineering at The Royal Institute of Technology, KTH, in cooperation with Tyréns AB.

First and foremost, I would like to give my sincerest gratitude to my supervisor and mentor Mahir Ülker-Kaustell, Ph.D. at KTH and Tyréns AB, for all the challenging and enlightening discussions, and for inspiring me towards further research within the field.

I would also like to give my thanks to Andreas Andersson, Ph.D. researcher at KTH, for the guidance and support throughout the thesis.

I am grateful for the opportunity that Tyréns AB has given me to write my thesis and I would specifically like to thank everyone at the bridge division for making it an enjoyable time. A special thanks to co-thesis writers Jossian Thomas and Assis Arañó Barenys for largely helping me with the bridge modelling.

Finally I want to thank my family, friends and loved ones for the support that they have given me throughout my five years of studying at KTH.

Stockholm, June 10, 2016

Johan Östlund

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Notations

Latin Letters

Notation Description

a

₀

Dimensionless coefficient

a

_n

Fourier series coefficient for cosine ex- pansion

b

_n

Fourier series coefficient for sine expan- sion

c Damping coefficient

c Viscous damping matrix

C

_d

Dymamic damping

c

_n

Fourier series coefficient for complex form expansion

DAF Dymnamic amplification factor

d Displacement

E Elastic modulus

E ¯ Complex elastic modulus

F Force in frequency domain

f Force in time domain

f

_D

Damping factor representing the rate- independent Kelvin solid

F

_n

Nyquist frequency

F

_R

Reaction force

F

_s

Sampling frequency

g Gravitational acceleration

H Frequency response function (FRF)

i Letter signalling imaginary numbers

I Moment of inertia

k Spring constant or stiffness coefficient

K

_d

Dymamic stiffness

k Stiffness matrix

K Diagonal stiffness matrix

L Length

m Mass in a SDOF problem

m Mass matrix

M Number of frequency domain samples

N Number of time domain samples

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n Integers in Fourier series etc.

n

₀

First eigen-frequency of a structure n

₂

First eigen-frequency of a structure n

_max

Maximum frequency included in dy-

namic analysis

p(t) Force

P

_j

DFT of signal

p

_n

IDFT of signal

Q Point load from HSLM analysis

T Total time of a signal

T

_n

Natural period of vibration

t Time

t

₀

Start time

U Displacement in frequency domain

u Displacement in time domain

U ˙ Velocity in frequency domain

˙u Velocity in time domain

U ¨ Acceleration in frequency domain

¨

u Acceleration in time domain

v

_design

Design speed of HSLM analysis

v

_LineSpeed

Speed limit on railway line

v

p

P-wave speed

v

_s

S-wave speed

w(t) Displacement

Greek Letters

Notation Description

γ Shear strains

ε Shear strains

ζ Damping ratio from Eurocode

η Loss factor

θ Angle of displacement

λ Wave length

ν Poisson’s Ratio

ρ Density

φ Frictional angle

φ

⁰⁰

Factor in calculating DAF

ω Circular frequency

ω

_n

Natural circular frequency

Ω Fundamental circular frequency

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Abbreviations

Abbreviation Description

2D Two-dimensional

3D Three-dimensional

AR Absorbing Region

BC Boundary condition

DAF Dynamic amplification factor

DFT Discrete Fourier Transform

DOF Degree of freedom

EC Eurocode

EOM Equation of motion

FE Finite element

FEM Finite element method

FEA Finite element analysis

FFT Fast Fourier Transform

GUI Graphical User Interface

HSR High Speed Railway

IFFT Inverse Fast Fourier Transform

OCR Over Consolidation Ratio

ODB Output database file

PML Perfectly matched layer

SDOF Single degree of freedom

SGI Swedish Geotechnical Institute

SSI Soil-structure interaction

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Chapter 1 Introduction

1.1 Background

High speed railways (HSR) are becoming an increasingly attractive investment for governments all over the world. The benefits are many; faster travels induce eco- nomic growth, the usage of trains instead of air planes or cars reduces the environ- mental impact, and the congestion and safety aspects as cars and car accidents may be reduced with less people on the roads. The Swedish Transport Administration (Trafikverket) is currently planning a new HSR link called Ostlänken between the cities of Stockholm and Linköping. This new HSR link will be stretching about 160 km and the maximum allowed train speed will be 320km/h (Trafikverket, 2016).

Ostlänken will in the future be linked up together with the planned Götalands- banan, which will be running between Gothenburg and Linköping. According to Trafikverket, this combined line will make it possible to travel between Stockholm and Gothenburg within 2 hours compared to the 3 hours of today (SJ, 2016). In the further future, the link will be extended to run to the city of Malmö, in the south of Sweden, and from there the plan is to connect it to Denmark and all the way down to the continent. Figure 1.1 shows a map of the south of Sweden where the new HSRs are prospected.

In the context of high speed rail planning and prospecting at present, railway bridges

with high demands on dynamic safety and cost-efficiency are in great need of design

and evaluation (Trafikverket, 2016). Investigations are in progress in the area of the

dynamic impact that high speed trains have on railway bridges. One type of railway

bridge that is under investigation is the end-frame bridge (also called end-shield

bridge). The end-frame bridge is a cost effective bridge type that is stabilized in

horizontal directions by the interaction between the embankment soil and thr end-

frames or wing walls. Figure 1.2 presents the end-frame bridge and its components.

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1.1. BACKGROUND

Figure 1.1: The planned HSR links in Sweden in blue. The first stretch that is under prospecting is planned to go between the cities of Stockholm and Linköping. Figure made by Andersson and Karoumi (2015).

Figure 1.2: Conceptual figure showing the end-frame bridge and its components.

Figure made by Ramic (2015).

Dynamic analyses on end-frame bridges with FE software based on shell and beam

theory neglecting the interaction with surrounding soil materials, has led to the

conclusion that this bridge type is not suitable for HSR. Today, research is conducted

at KTH Division of Structural Engineering and Bridges which indicates that end-

frame bridges’ interaction with embankments has a significant and favorable effect

on the dynamic properties of the bridge type (Andersson et al., 2010).

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1.2. STANDARDS

line in the north of Sweden. The aim of the field tests were to establish the dynamic response and dynamic properties of the bridge type from one that is actually in operation in reality in order to validate the theoretical results. The field tests were conducted with a hydraulic bridge exciter, the same method as applied by Andersson et al. (2015).

Figure 1.3: End-frame bridge Aspan on the Bothnia rail link. (BaTMan, 2015)

1.2 Standards

The Swedish Transport Administration requires a dynamic analysis on railway bridges in addition to the static for railway lines with speed limits above 200km/h (TRVK Bro, 2011). The main difference between static and dynamic analysis is that the effects of resonance is considered. For HSR, it is common that the design is gov- erned by the requirements on dynamic response of bridges. The speed limit on the planned HSR is 320km/h and thus a dynamic analysis is required. In this chapter, a brief description of the demands that applies on dynamic analysis of railway bridges according to Eurocode is given.

The following bridge responses must be checked when a dynamic analysis is required SS-EN 1990 (2002):

• Vertical bridge deck acceleration

• Vertical and horizontal displacements

• Rotations at bearings and supports

• Torsions

1.2.1 Standards and Documents

Standards and design criteria that govern bridge design in BLABLABLA are pre-

sented in this section.

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1.2. STANDARDS

Eurocode is the basis of all structural and geotechnical design in Sweden. Eurocode allows for specific national regulations in national annexes at the end of each stan- dard. In this thesis, Eurocode SS-EN 1990 and SS-EN 1991-2 have been frequently referred to, but other standards have also been followed.

TRVK Bro 11 is the Swedish Bridge Standard and is maintained by the Swedish Transport Administration.

TK Geo 13 Krav is a standard maintained by the Swedish Transport Administra- tion as demands on geotechnical design. TK Geo 13 Råd is also produced by the Swedish Transport Administration and gives advice on geotechnical design matters.

TK Geo 13 Krav and TK Geo 13 Råd complement each other and have the same section numbering. The samegoes for TRVK Bro(TRVR).

SGI (Swedish Geotechnical Institute) releases information on geotechnical material properties of soil and gives advice on appropriate actions at specified environmental conditions. In this thesis, SGI-i1 and SGI-i17 (information 1 and 17) are frequently used. SGI-i1 is a general information on material properties in geotechnics. SGI-i17 gives advice on dynamic soil properties.

1.2.2 Dynamic analysis

In this section, he dynamic analysis of bridges for HSR is presented. This section has been subdivided and presented in three parts in this thesis; one for the bridge response limits such as accelerations and displacements. The second part states requirements regarding loading. The third part presents bridge input parameters such as damping etc. Finally, a short summary of the regulations applicable to this thesis is presented.

Bridge response limits

The following limits on the bridge response that must be evaluated in a dynamic analysis are stated in SS-EN 1990, section A2.4.4.

According to section A2.4.4.2.1(4), the vertical accelerations should be less than 5 m/s

²

for un-ballasted tracks. The corresponding value for a ballasted track is 3.5 m/s

²

which is set to avoid ballast instability. The un-ballasted tracks restriction is set to avoid de-railing and to maintain traveller comfort.

The maximum frequency that is included in the analysis should be limited to the

maximum of one of the following frequencies: n

_max

= max(30Hz, 1.5×n

₀

, n

₂

), where

n

₀

is the first eigen-frequency of the structure and n

₂

is the third eigen-frequency of

the structure. In this thesis, 30 Hz has been used as the maximum frequency in the

analysis.

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1.2. STANDARDS

the acceleration has, instead, been translated into a requirement on the maximum displacement. The limit on displacements are dependent on the longest span length and the train speed and limits may be obtained through figure A2.3 in the Eurocode, see figure 1.4. For simply supported beams with one or two spans, and for continuous bridges with two spans, the results from figure A2.3 should be multiplied by a factor of 0.7. For continuous bridges with three spans or more, the results should be multiplied by 0.9.

Figure 1.4: Figure A2.3. in SS-EN 1990. Limits of the maximum displacement of the bridge may be obtained through the determinant span length and the speed of the train.

Rotations at supports and torsions have not been considered within this thesis. They must, however, be verified in a real design process.

Loading

The following section presents some basic features regarding the load model that should be used when performing dynamic analysis according to EC. The require- ments on loading are stated in SS-EN 1991-2 chapter 6.4.6.

The HSLM load model is subdivided into HSLM-A and HSLM-B. HSLM-B com-

prises a number of point forces with uniform spacing, while HSLM-A is designed to

resemble real trains. The HSLM-A is shown in figure 1.5. It consists of 10 universal

train types that represent all kinds of trains that would be likely to run on the

railway. According to table 6.4 in the Eurocode, for spans longer than 7 meters, the

HSLM-A analysis must be run with all 10 universal train types.

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1.2. STANDARDS

Figure 1.5: HSLM-A, figure 6.12 in SS-EN 1991-2, shows a conceptual representation of the train load model.

Each wheel load may be distributed in the longitudinal direction as point loads on sleepers as figure 1.6 shows.

Figure 1.6: The longitudinal distribution of each axle load on rails. Figure 6.4 in SS-EN 1991-2

For double-tracked bridges, only one of the tracks at a time is required to undertake dynamic loading, according to table 6.5 in SS-EN 1991-2.

For each HSLM load model a series of speeds from 40 m/s (144km/h) up to the maximum design speed should be analyzed according to section 6.4.6.2. The design speed should be taken as equal to the maximum allowed speed on the track, times 1.2.

v

_design

= v

_{line speed}

× 1.2 (1.1)

Bridge parameters

The structural damping,ζ , of the bridge should be chosen according to table 6.6

(here, figure 1.7). This is a lower limit value that may be applied and is estimated

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1.2. STANDARDS

Figure 1.7: Minimum values of the structural damping of bridges that may be as- sumed. The values are dependent on the bridge type and the determi- nant span length. (SS-EN 1991-2, 2003)

According to section 6.4.6.4(4) in SS-EN 1991-2, additional structural damping may be added to the damping in figure 1.7 for spans less than thirty meters due to vehicle-bridge mass interaction. However, no such damping has been added to this thesis.

The dynamic response should be multiplied by a Dynamic Amplification Factor (DAF) equal to 1 + 0.5φ

⁰⁰

according to section 6.4.6.4(5) for carefully maintained tracks. This DAF takes track defects and vehicle imperfections into consideration and is calculated from the maximum permitted speed, the first natural bending frequency of the bridge, and a characteristic length of the bridge. φ

⁰⁰

is calculated according to Annex C in SS-EN 1991-2.

End-frame bridges

There are neither requirements nor design suggestions that directly aim at taking dynamic response of end-frames and soil structure interaction into consideration in either SS-EN 1990 or SS-EN 1991-2.

According to SS-EN 1992-2, section 4.9.1, the carriageway located behind abut- ments, such as wing walls, side walls and other parts of the bridge in contact with soil should be loaded with "appropriate model in vertical direction", meaning a load corresponding to the vehicle load. For vertical walls, e.g. end-frame walls, corre- sponding breaking forces in horizontal direction should be applied. These demands are for static evaluation and does not take dynamic effects into consideration. In SS-EN 1990 section A2.3, there are suggestions on how to apply loads on structural elements in contact to soil. Again, this is for static design.

In SS-EN 1990, section 5.1.3(2) it is stated that "The boundary conditions applied to the model shall be representative of those intended in the structure.". Furthermore, in section 5.1.3(4) it is stated that "In the case of ground-structure interaction, the contribution of the soil may be modelled by appropriate equivalent springs and dash-pots".

This may be interpreted such that soil-structure interaction must be included in the

analysis when a structure and surrounding soil is integrated. However, the statement

does not say how the springs and dash-pots may be obtained.

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1.3. THE MAIN ISSUE

1.3 The main issue

The dynamic behavior of end-frame bridges are dependent on the soil structure interaction (SSI) between the end-frame and the soil. As it is currently not specified in the Eurocode how the SSI of end-frames can be included, the favorable effects of SSI cannot easily be accounted for in theoretical modelling.

Since high-speed traffic may induce considerable dynamic effect on bridges, this has led to the judgement that end-frame bridges should not be used for high speed rail- ways as there is a lack of prediction on the effectiveness of end-frames. The task of this thesis has been to provide a way to utilize the interaction and, in the end, simplify the interaction to functions of springs and dash-pots that will be attached to the bridge. The SSI functions consisting of springs and dash-pots are called impedance functions and may represent the SSI in translations as well as rotations.

Figure 1.8 visualizes the issue of end-frame bridges when SSI is not included in the model as a 2D-representation. (a) shows the end-frame bridge as a consoling simply supported beam. The effects of SSI is not included. (b) shows that as the train arrives at the bridge, the lack of support causes large vibrations. In (c), the bridge is connected to impedance functions, consisting of springs and dash-pots, acting on translational as well as rotational DOFs.

Figure 1.8: 2D-representation of the main issue of the problem when not including

SSI on end-frame bridges.

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1.4. AIMS AND SCOPE

1.4 Aims and scope

The aims of this thesis were subdivided into three parts, stated below:

1. Perform a parameter study on the impedance of the interaction between end- frames and the soil.

2. Perform dynamic HSLM analyses on bridges including effects of impedance functions gained in step 1.

3. Analyse test data from field tests on bridge Aspan and compare to modelled

results gained in step 2.

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Chapter 2 Theoretical background

In dynamic modelling within the field of civil engineering, the dynamic aspects are necessary to evaluate whenever the conditions of a structure or foundation are changing critically in time. Dynamic loading may induce vibrations on a structure which at critical frequencies cause the structure or foundation to enter resonance state. Some dynamic loads that act on bridges that may cause resonance on bridges are stated:

• Vehicles such as trains, cars and trucks

• Pedestrian loading

• Wind loads

• Earthquakes and other movements of the foundation or soil.

• Hydraulic loading such as flowing water.

• Blast and impulse loads

When modelling dynamic loading in time domain in FE software it is necessary to subdivide time into time steps. It is important to set a short enough time step in order to simulate properly without missing important variations in the load function.

However, choosing a too short time step is computationally costly meaning that the time that is needed for solving the model in FE software becomes too large. The time domain approach is usually a relatively computationally costly method for gaining the dynamic response of a structure due to the need of small time steps.

An alternative method of gaining results from dynamic problems is the frequency

domain approach. By the use of Fourier transform one utilizes the opportunity

to shift between time and frequency domain and with the fast Fourier transform

(FFT) algorithm (Cooley and Tukey, 1965) it is today possible to attain accurate

results within short computational time. The frequency domain approach to solving

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2.1. FREQUENCY DOMAIN APPROACH

Generally when working with SSI, impedance is an important concept. Impedance functions represent the dynamic stiffness and damping of soil. The damping consists of two parts; material damping and radiation damping, generated from energy car- ried away in the soil from the foundation/structure in form of waves. The concept of impedance functions is described in section 2.2.

To obtain correct impedance functions from modelling, it is important to fulfil the radiation condition in order to simulate wave propagation correctly. In this thesis, linear elastic waves have been assumed. This will be further explained in section 2.3.

To avoid computationally costly models, the use of linear elastic soil material models may be used. To ensure that the soil affected by wave motion in fact is linear elastic, one must check the limitations of the constitutive model. It is assumed that for soils, this condition is fulfilled for shear strains less than 10

⁻

4.The theory of the constitutive model in terms of stiffness and damping in the frequency domain is further described in section 2.4.

Finally, the last section will include the design of an absorbing region. It is necessary to restrain the model in order to avoid an unmanageable number of degrees of freedom. If the assumption is that there is no natural boundary in reality, and the model must resemble reality, the boundary of the calculation domain (the actual model) must have an absorbing boundary which does not allow for reflections that may return to the source (structure/foundation). In this thesis, an absorbing region (AR) has been used. More about its design and reasoning behind the choice of absorbing boundary may be read in section 2.5.

2.1 Frequency domain approach

The frequency domain approach to solving dynamic problems is presented in this section. First, the equation of motion is presented. Secondly, the Fourier transform and applications of it used in this thesis are briefly described. Thirdly, the steady- state dynamic method is derived.

EOM in time domain

In this section, the equation of motion (EOM) is first defined in time domain and later on the Fourier transform derives the EOM in frequency domain. This section is based on derivations made by Chopra (2014).

Consider the single degree of freedom (SDOF) problem in figure 2.1.

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2.1. FREQUENCY DOMAIN APPROACH

Figure 2.1: EOM may be derived from a spring mass damper system such as in this figure.

The mass body acts with a resultant equal to the applied load and inertia force.

P (t) = F (t) − m¨ u(t) (2.1)

The resistance resultant depends on the spring stiffness and dash-pot damping as:

P (t) = c ˙u(t) + ku(t) (2.2)

If eq. 2.1 and eq. 2.1 is put equal to each other the equation of motion (EOM) is received:

m¨ u(t) + c ˙u(t) + ku(t) = F (t) (2.3)

EOM in frequency domain

The Fourier transform of a function in frequency domain is expressed as:

F (ω)e

^iωt

(2.4)

The complex solution in the differential equation may be expressed as:

u(t) = U (ω)e

^iωt

(2.5)

Similarly, the derivate of eq. 2.5 leads to

˙u(t) = iωU (ω)e

^iωt

(2.6)

˙u(t) = −ω

²

U (ω)e

^iωt

(2.7)

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2.1. FREQUENCY DOMAIN APPROACH

2.1.1 Fourier Transform

This section gives a short presentation of the Fourier transform and is based on derivation made by Chopra (2014) and Vretblad (2003).

The Fourier series may be used to represent arbitrary periodic functions as series of trigonometric functions. The Fourier transform is used to decompose signals in the time domain into its constituent frequencies in frequency domain, and is based on the Fourier series. Assuming a linear change of variable, the Fourier series can be expressed as:

f (t) ∼ ^X

n∈Z

c

n

e

^inΩt

where c

_n

= 1

2T

Z

T

−T

f (t)e

^−inΩt

dt

(2.9)

Or, alternatively,

f (t) ∼ 1 2 a

₀

+

∞

X

n=1

(a

_n

cos nΩt + b

_n

sin nΩt) (2.10) where,

a

n

= 1 T

Z

T

−T

f (t) cos nΩtdt, n = 0, 1, 2, ...

b

_n

= 1 T

Z

T

−T

f (t) sin nΩtdt, n = 0, 1, 2, ...

(2.11)

Here, Ω may be called the fundamental angular frequency and T is the period.

The Fourier series represent periodic functions. Figure 2.2 shows the relationships

between time domain and frequency domain. The figure shows how the Fourier

series may subdivide the original signal into two fundamental signals with their own

amplitudes and frequencies.

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2.1. FREQUENCY DOMAIN APPROACH

Figure 2.2: The relationship between time domain and frequency domain. a) shows how a signal may be plotted in both time and frequency domain at the same time, b) is the time domain response, and c) is the frequency domain response. (Hewlett-Packard, 2000)

The Fourier series may be generalized into the Fourier integral which has the advan- tage of being able to represent non-periodic functions. The complex Fourier integral is given in eq. 2.12.

f (t) = 1 2T

Z

∞

−∞

F (ω)e

^iωt

dt (2.12)

This equation is also referred to as the inverse Fourier transform of the frequency de- pendent function F (ω). The Fourier transform, also called direct Fourier transform is given as

F (ω) =

Z

∞

−∞

f (t)e

^−iωt

dt (2.13)

The response to an arbitrary excitation of a linear system is received by combining responses to individual harmonic excitations in terms of Fourier integrals, as in eq.

2.12. Assume an excitation P (ω)e

^iωt

. The response of the system is given by:

U (ω) = H(ω)P (ω)e

^iωt

(2.14)

If eq. 2.14 is inserted into eq. 2.12 the response in time domain is obtained as u(t) = 1

2π

Z

∞

−∞

U (ω)P (ω)e

^iωt

dt (2.15)

In signal analysis and in FE modelling the values are sampled digitally in a discrete

manor. As the Fourier integrals are continuous functions it is necessary to adapt

them for discrete sampling. The discrete Fourier transform (DFT) is, in contrast

to the continuous, a numerical evaluation. The numerical evaluation demands trun-

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2.1. FREQUENCY DOMAIN APPROACH

Fourier transform is a true representation of the excitation function over infinite range, the discrete transform only represents a periodic version of the function.

The DFT pairs are expressed as:

p

_n

(t) =

M

X

j=−M

P

_j

(ω)e

−i(2πnj/N )

(2.16)

P

_j

(ω) = 1 N

N −1

X

n=0

p

_n

e

−i(2πnj/N )

(2.17) Where p

_n

is a discrete array that represents the continuous function p(t) as a superposition of N harmonic functions. N is also the number of equally spaced samples equal to the total sampling time T

₀

divided by the sampling interval ∆t as:

N = T

₀

∆t (2.18)

Choosing ∆t sufficiently small, ensures accurate representation of the force excita- tion as well as the forced vibrational component of the response. By choosing T

₀

sufficiently large ensures accurate representation of the free vibrational component.

The DFT is called a one-sided Fourier expansion since it does only consider positive frequencies (see eq. 2.17).The continuous Fourier transform is a two-sided expansion since it considers both positive and negative frequencies. Negative frequencies have no physical meaning in a two-sided expansion. This, however, also applies to fre- quencies corresponding to N/2 < j ≤ N − 1, which are counterparts to the negative frequencies.

The Nyquist frequency or folding frequency:

f

_max

= f

_{N yquist}

= N

2 ω

₀

= N

2 2πf

₀

= π

∆t (2.19)

where f

_o

is the fundamental or first harmonic in the excitation as:

f

₀

= 1

T

₀

(2.20)

The periods of of the harmonics that is included in a Fourier expansion should be from 2∆t to T

₀

in length. As stated before, DFT is a periodic interpretation of the original function, and as such, it also gives the best results if the values at the first and last time steps are zero valued. If this is not the case, response from one side of the function, or signal, will be reflected to the other side causing erroneous transforms.

The method of using DFT to determine dynamic responses was practically made

possible as Cooley and Tukey (1965) developed an algorithm to efficiently compute

the DFT without extensive computational cost. This new algorithm is known as the

fast Fourier transform (FFT) and algorithms based on the Cooley-Tukey algorithm

are widely used in software such as MATLAB or Python (Numpy).

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2.1. FREQUENCY DOMAIN APPROACH

Zero-Padding

Zero-Padding is used to increase the number of frequency steps and the maximum frequency in frequency domain. By increasing the number of frequency steps, the time domain response, from the inverse fast Fourier transform (IFFT), becomes more accurate in terms of making the time steps smaller.

One-sided and double sided

From equation 2.16 and 2.17 one may see that the series in frequency domain only have positive samples, as n=0 to N-1. This is also called a one-sided series. On the other hand, the time domain series has its samples from negative M to positive M. This series may also be called double-sided. Negative frequencies are not real and when calculating the frequency response in a model, only responses for positive frequencies are obtained. However, when an IFFT is performed, it is necessary to adjust the response into a double-sided function. Since the frequency response consists of complex values, both imaginary and real numbers must be considered.

The double-sided function is achieved by mirroring the complex conjugate, meaning that the real part is simply mirrored, while the imaginary part is mirrored with a negative sign.

Windowing function

In this thesis the Tukey window function has been used to filter away real and imaginary numbers in frequency domain that are at the end of the frequency intervals (negative and positive) which are to be zero-padded from the maximum frequency.

By filtering away these responses, one avoids a discontinuity in frequency domain that may disturb the results in time domain gained from IFFT.

Sampling frequency intervals

The frequency response is greatly increased at states of resonance and in between resonance peaks the response is less drastic. To save computational time, it therefore would be beneficiary to not have the same sampling frequency interval in between peaks and to have an increased sampling frequency interval where peaks are located.

The method to do this is called a biased frequency sampling and may be differently adjusted in different FE-software, for example. As resonance occurs at eigen fre- quencies, the eigen frequencies need to be known in order to be sure where to expect resonance peaks.

In the standard DFT, the frequency samples are evenly distributed between the

maximum and minimum frequencies of interest, that is, that the sampling frequency

interval is constant. The biased samples must therefore be interpolated with an even

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2.2. IMPEDANCE

2.1.2 Steady-state dynamics

This section about steady-state analysis is based on derivations from Chopra (2014) and (Vretblad, 2003).

If the load p(t) is put equal to the unit harmonic load:

p(t) = 1e

^iωt

(2.21)

Then, for any system, the steady-state response of is a harmonic motion expressed as:

u(t) = ¯ H

_u

(ω)e

^iωt

(2.22)

Where ¯ H

_u

(ω) is the frequency response function (FRF). If eq. 2.21 and 2.22 is put into eq. 2.8 EOM is given in the steady-state:

h − ω

²

m + iωc + k ⁱ H ¯

u

(ω) = 1 (2.23) The damping coefficient c is chosen to be a rate independent (see section 2.4) as:

f

_D

= ηk

ω ˙u (2.24)

where, η is the loss factor. Eq. 2.24 put into eq. 2.23 yields:

h − ω

²

m + k(1 + iη) ⁱ H ¯

u

(ω) = 1 (2.25) And finally, the FRF may be explicitly expressed as

H ¯

u

(ω) = 1

h − ω

²

m + k(1 + iη) ⁱ (2.26) The stiffness and the rate independent damping dependent on the stiffness can be explained by the rate independent Kelvin solid. If the stiffness is substituted by Young’s modulus the complex modulus is obtained:

E = E(1 + iη) = E(1 + i2D) ¯ (2.27) Where η is equal to two times the equivalent viscous damping ratio, 2D.

2.2 Impedance

Impedance functions represent the dynamic stiffness and damping of a soil-structure

interface. The damping consists of two parts; material damping, energy losses due to

hysteretic action in the soil and radiation damping, generated from energy carried

away from the source, e.g. foundation/structure in form of elastic waves. The

impedance function is inversely proportional to the receptance. The receptance is

the FRF in frequency domain at the same point as a load is applied. This section is

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2.2. IMPEDANCE

dedicated to the derivation of impedance functions from EOM and the methodology of acquiring impedance functions based on concepts stated by Gazetas (1990).

The denominator in eq. 2.26 is replaced by a frequency dependent function ¯ K(ω) giving

H ¯

_u

(ω) = 1

h − ω

²

m + k(1 + iη) ⁱ = 1

K(ω) ¯ (2.28)

K(ω) is called the impedance function and may be separated into two parts, a real ¯ and an imaginary part.

K(ω) = K ¯

_d

(ω) + iωC

_d

(ω) (2.29) The FRF that is dependent on the impedance functions is called the receptance.

The receptance may be expressed as:

H ¯

u

(ω) = 1

K

_d

(ω) + iωC

_d

(ω) (2.30)

A conceptual figure of soil-structure interaction and the concept of interface impedance

is presented in figure 2.3. In the figure, a dynamic force is applied to a structure

or foundation positioned on top of soil. The structure is affected by the force, the

inertia force and the soil reaction resultant. The soil reaction resultant force consists

of damping as well as stiffness and is, as such, dependent on the displacement as

well as the speed of the vibrating structure. As the structure vibrates, it will start

to initiate wave motion in the soil. These waves transfers energy from the structure

and away in the soil and, in that way, damps the vibrations in the structure. The

impedance includes thus both the dynamic stiffness and dynamic damping of the

soil. The material of the structure has been assumed infinitely stiff compared to the

soil, meaning that the structure will displace the material equally distributed along

its surface interface. This induces the stress distribution with higher stresses at the

edges of the structure, visible in the figure. This is called interface impedance.

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2.2. IMPEDANCE

Figure 2.3: A representation of the interface impedance of SSI between structure and the soil. (Gazetas, 1990)

It is important to include responses of all translational and rotational directions at

the particular node, even if the load is only applied in one direction. Rocking is one

example where a rotational load creates a movement both rotationally (the direction

of the load) and translationally (the edges of e.g. a plate goes up and down with a

sectoral circular pattern). The rocking behaviour is visualized in figure 2.4.

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2.2. IMPEDANCE

Figure 2.4: Figure showing how an applied dynamic moment may put the structure into a rocking movement.

As the impedance is made up from the inverted receptance, which is a 6x6 matrix, all terms in the matrix is part of the inversion. It may be argued that only the diagonals of the impedances will play a role in the end, as the other terms will typically be much smaller compared to the diagonals.

2.2.1 Calculating impedance functions

When computing theoretical impedance functions, there are some conditions that must be fulfilled in order to obtain accurate results.

The radiation condition

To be able to model waves that origins from the foundation or structure accurately

it is necessary to know the different wave types and how they act in elastic half-

spaces. By considering the relevant wavelengths the mesh size and model length

may be specified. The boundaries which limits the half-space must be wisely cho-

sen to resemble reality, both in terms of restraints but also in terms of absorbing

boundaries. The absorbing boundaries are used to resemble a continuation of the

model

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2.3. WAVE MOTION IN ELASTIC SOLIDS

Constitutive model

By using linear elastic, homogenous, and isotropic material computational costs may be reduced compared to running models including non-linear material properties.

However, linear elasticity is, in general, only valid for very small strains, particularly for soils. Therefore, there is a need to check shear strains in the model to make sure that no strains exceed the limit value that is the difference between linear and plasto-linear behavior.

Interaction of end-frame

The interaction between the end-frame and the soil is dependent on the stiffness of the concrete, stiffness of the soil and the friction that may be assumed between them. Since soil typically has a much lower modulus than that of concrete, it has been assumed that the concrete surfaces in contact with the soil is infinitely stiff compared to the soil. Furthermore, the friction between them has been assumed to be 100%. This might not be entirely true, which will be argued in the discussion, section 5.2. The assumption of 100% friction implies very high stress levels in soil elements close to the interface, since they will be restrained in all translational and rotational directions.

2.3 Wave motion in elastic solids

This section about wave motion in elastic solids is based on the books by Graff (1975), Kausel (2005) as well as the geotechnical standard SGI-i17 (2008). There are four major types of waves in elastic solids. The P-wave, or compressional wave, has a soil motion which is parallel to the direction of the wave. S-waves, or shear wave, on the other hand, has soil movements that are perpendicular to the direction of the wave. Rayleigh and Love waves are both surface waves. The Rayleigh wave has an elliptical soil motion in planes normal to the surface and parallel to the direction of the wave. It may be compared to surface waves on water. Love waves may only occur when there are layered materials with different stiffness, and in particular when a material layer has a free surface on one side and a stiffer layer on the other. The magnitude of both of these surface waves are reduced by depth.

P-wave is a shortening of primary wave and comes from the fact that it is the fastest

wave, and thus is the first wave that will reach a certain distance. S-wave stands

for secondary wave, since it reaches second. Love waves are faster than Rayleigh

waves, which is the slowest wave type. In saturated soil, the P-wave enters the

speed of water, which is usually higher than that of the soil. The uncertainties of

the measured speed is why in geotechnics this wave is not used in order to decide

geotechnical properties. In contrast to the P-wave, the S-wave does not travel in

water (shear waves do not exist in water) and does only travel through the grains,

the skeleton, of the soil. It is therefore used in geotechnics to decide geotechnical

properties of soil. The wave types are visually presented in figures 2.5, 2.6, 2.7 and

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2.3. WAVE MOTION IN ELASTIC SOLIDS

2.8. The P-wave speed may be calculated according to the following relationship:

v

p

= M

_c

ρ (2.31)

Where ρ is the density and M

_c

is the compression modulus, related to Young’s modulus as:

M

_c

= E(1 − ν)

(1 + ν)(1 − 2ν) (2.32)

The shear wave speed may be calculated from the shear modulus:

v

s

= G

ρ (2.33)

The shear modulus is related to Young’s modulus according to:

G = E

2(1 + ν) (2.34)

To calculate a wavelength for a specified frequency and wave speed, the following formula may be used:

λ = v

_i

f (2.35)

The P-wave travels in the speed of water in saturated soil, and as water has a higher speed than soil in general, the P-wave rather reflects the speed of waves in water instead of the soil speed.

Figure 2.5: The P-wave, also compression wave, has its soil motion in the same

direction as the wave. It is the fastest of the wave types, hence the P

(Primary) in P-wave.

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2.3. WAVE MOTION IN ELASTIC SOLIDS

Figure 2.6: The S-wave, also shear wave, has its soil motion perpendicular to the direction of the wave. It is the second fastest wave (Secondary).

Figure 2.7: The Rayleigh wave is a surface wave with a circular soil motion. This wave type is hat may be observed as the surface waves on water.

Figure 2.8: The Love wave is a surface wave that only occurs when a stiff soil layer is underneath a soft soil layer at the surface of the soil.

Reflections and refractions

As waves travel in a soil layer, with material stiffness and damping, and hits the

interface of another material layer, with different stiffness and damping, the wave is

affected (Kausel, 2005). The phase velocity of the wave is changed but the frequency

remains constant. The way that the wave is affected is dependent on the angle that

the wave hits the interface with, and also dependent on the difference in stiffness

and damping.

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2.4. SOIL MATERIAL MODELS

If a wave travels in one soil layer, and hits a layer with infinite stiffness the wave will be reflected against the surface. A boundary that is fixed may be interpreted as an infinitely stiff material.

The difference of material properties between two layers causes always one part of the wave to be refracted with a small change of angle compared to the original direction of the wave, while the other part is reflected backwards. Therefore, it should be avoided to have a too big change in material in a model, where, in reality, the distribution of material properties vary from point to point, and not in layers.

2.4 Soil material models

Soil is a non-linear material. Figure 2.9 shows a schematic plot over the non-linearity of soil. For high strains in the soil, a linear elastic model may not be used. However, it has been assumed that shear strains for this SSI problem are typically very low.

Figure 2.9: The non-linearity of soil material. τ is shear stress, γ is shear strain and G is the secant modulus. G

₀

is the maximum secant modulus, valid at very low strains (SGI-i1, 2000).

Shear modulus

Extensive research on the shear modulus and damping of soils was conducted by

Hardin and Drnevich (1972) and that research is still today the basis of modern soil

dynamics. Several researchers later on have been using similar empirical models as

were stated by the two authors and have been aiming to simplify them for specific

cases. The work Hardin and Drnevich (1972) engaged in was to determine shear

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2.4. SOIL MATERIAL MODELS

damping and moduli is:

• Strain amplitude, γ

• Effective mean principal stress, σ

_m⁰

• Void ratio, e

• Number of load cycles

• Degree of saturation (cohesive soils) Secondary factors were suggested to be:

• Octahedral shear stress

• Over Consolidation Ratio (OCR)

• Effective stress strength parameters, such as cohesive strength and frictional angle

• Time effects

There are several formulas to chose from, made by Hardin and Drnevich or modified by more recent authors, that are more or less suitable for different soil types and cases. The chosen formulas will be presented and discussed in the Method chapter, see section 3.1.1.

Shear strains

Shear modulus of soil is largely dependent on shear strain. In figure 2.10, presented

by Ishihara (1996), the limits for different behaviors of soil at different strain levels

is presented. The author suggests that for strains less than 10

⁻⁴

(= 0.01%) elastic

soil models may be used (preferably for strains less than 10

⁻⁵

). At medium strain

range, the soil becomes elasto-plastic and the shear modulus decreases in magnitude,

see figure 2.9. While the modulus decreases, the damping due to energy dissipation

in each cycle increases. Ishihara (1996) claims that energy dissipation in soil is

mostly rate-independent and due to hysteretic action, and that the damping ratio

may represent the energy absorbing behavior of such material. At medium strain

level, the soil modulus and damping is not affected by effects of load repetition and

loading rate. This material phenomenon is called non-degraded hysteresis type and

may be well enough described by visco-elastic theory, which is valid for strains less

than 10

⁻⁴

. The rate-independent but strain dependent behaviors of the soil may

be analyzed using the equivalent linear method based on visco-elastic theory. At

large strain levels, effects of load repetition and loading rate become evident. This

phenomenon is called degraded hysteresis type. Finally, the large shear strain in the

Failure strain level causes the soil to fail.

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2.4. SOIL MATERIAL MODELS

Figure 2.10: This figure shows soil behaviors dependent on shear strain level. In this thesis, the viscoelastic model has been used for strains less than 10

⁻⁴

. (Ishihara, 1996)

Maximum Shear Modulus

The maximum shear modulus, or the initial static shear modulus, may be calculated from the average principal stress at one point according to:

σ

₀⁰

= 1 + 2K

₀

3 σ

⁰_v

(2.36)

where

σ

_v⁰

= σ

_v

− u (2.37)

is the effective vertical stress component equal to the vertical stress minus the pore water pressure (here, assumed to be zero in the embankment). K

₀

is the at-rest earth pressure coefficient dependent on the drained frictional angle. This empirical relationship was first estimated by Jaky (1944) and is expressed as:

K

₀

= 1 − tan φ

⁰

(2.38)

This relationship is valid for frictional soils. Mayne and Kulhawy (1982) suggested another empirical relationship where also the Over Consolidation Ratio (OCR) is included:

K

₀

= (1 − tan φ

⁰

)(OCR)

^φ⁰

(2.39) This relationship is valid for soil ranging from clay to gravel. There are other empirical relationships that strive to explain the lateral earth pressure coefficient.

In this thesis however, eq. 2.4 has been used. Eq. 2.4 is presented to show that

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2.4. SOIL MATERIAL MODELS

may be calculated as a function of the square root of the average principal stress and three functions depending on OCR, void ratio (e), and plasticity index (I

P

).

G

_max

= f (OCR) f (e) f (I

_P

) ^q σ

₀⁰

(2.40) This, too, is an empirical relationship that depends on a large number of parameters.

Visco-elastic model

This section about soil material models are based on books by Kramer (1996) and Ishihara (1996). Hysteretic damping, sometimes also called structural damping.

A sinusoidal shear stress is applied to a body with an viscoelastic response. The shear stress is

τ = τ

_a

sin ωt (2.41)

where τ

_a

is the amplitude, ω is the circular frequency and t is the time. The vis- coelastic response to the applied shear force will have a small time delay, δ, as

γ = γ

_a

sin ωt − δ (2.42)

Similarly, a cosine stress induces a cosine response with a time delay. By introducing the complex variables of the stress and strain respectively, and expressing the sine and cosine parts in terms of real and imaginary part, the following equations are obtained;

¯

τ = τ

_a

e

^iωt

¯

γ = γ

_a

e

^i(ωt−δ)

(2.43)

The stress-strain response is gained thru division of the stress by the strain as;

¯ τ

¯ γ = τ

_a

γ

_a

e

^iδ

= τ

_a

γ

_a

(cos δ + i sin δ) (2.44) By putting

µ = τ

_a

γ

_a

cos δ, µ

⁰

= τ

_a

γ

_a

sin δ, µ

^∗

= µ + iµ

⁰

(2.45) From eq. 2.4, eq. 2.4 may be simplified and rewritten as

¯ τ

¯

γ = µ + iµ

⁰

= µ

^∗

(2.46)

µ is called the elastic modulus and µ

⁰

is the loss modulus. µ

^∗

is called the complex modulus. The formulas in eq. 2.4 may be rewritten as the absolute value of the complex modulus as

τ

_a

γ

_a

= ^q µ

²

+ µ

⁰²

= |µ

^∗

| , tan δ = µ

⁰

µ = η (2.47)

η is called the loss factor and may also be defined in terms of damping ratio as

η = 2ξ (2.48)

Where ξ is the damping ratio.

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2.4. SOIL MATERIAL MODELS

Hysteresis loop

In figure 2.11, the hysteresis stress-strain curve is presented. Each loop on the ellipse represents one cycle of loading and the area each load cycle creates (as in figure 2.11) represents the energy losses. The energy dissipation of one cycle of harmonic vibration caused by rate independent viscous damping may be expressed as

E

_D

= πηku

²₀

= 2πηE

_So

(2.49)

Where u

₀

is the amplitude of the motion and E

_So

= ku

²₀

2 which represents the elastic strain energy. Thus, ∆W represents the energy losses due to damping. The elastic strain energy in the soil is represented by the area W.

∆W =

Z

τ dγ = µ

⁰

πγ

_a²

, W = 1

2 µγ

_a²

(2.50)

From this, the loss coefficient, first defined in eq. 2.4, is defined from the hysteresis loop areas as:

η = 1 2π

∆W W = µ

⁰

µ = tan δ (2.51)

Figure 2.11: The hysteresis loop of the linear visco-elastic model. The triangular

area represents the elastic energy conserved in each load cycle. The

dotted area represents energy losses due to hysteretic (material) damp-

ing for each load cycle. (Ishihara, 1996)

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2.4. SOIL MATERIAL MODELS

Rate-dependent Kelvin solid

The Kelvin model is a simple and widely used constitutional model based on the concept of a spring and a dash-pot connected in parallel. Here, the strain, γ, is impelled to the two elements equally, while the stress is divided into two parts where one is carried by the spring, τ

_spring

, and the other by the dash-pot, τ

_dash−pot

, and added together as τ = τ

_spring

+ τ

_dash−pot

. The stress that is distributed to the spring may be expressed as Gγ and the dashpot as G

^{0 dγ}_dt

.The total stress is then given as,

τ = Gγ + G

⁰

dγ

dt (2.52)

The complex variant of stress and strains are, as in eq. 2.4 is put into eq. 2.4 giving τ

_a

e

^iδ

= (G + iωG

⁰

)γ

_a

(2.53) The relationships in eq. 2.4 is used to rewrite eq. 2.4 to

µ + iµ

⁰

= G + iωG

⁰

(2.54)

where

µ = G, µ

⁰

= iωG

⁰

, η = tan δ = ωG

⁰

G (2.55)

where η is the loss coefficient. If eq. 2.4 together with the relationships in eq. 2.4 is rearranged, the following relationship is obtained

G = (1 + iωη)G ¯ (2.56)

Rate-independent Kelvin solid

The damping of a rate-dependent Kelvin solid, based on visco-elastic theory, in- creases naturally with the frequency as it origins from models of dashpots where damping is velocity dependent. Soil, however, does not have an increase of damping due to an increase of frequency but is typically frequency dependent. From this, a modification of the rate-dependent model is needed. The rate-independent Kelvin model is defined as

τ = (G + iG

⁰₀

)γ (2.57)

Where G

⁰₀

is a dashpot constant. Compared to eq. 2.4, ω has been removed and G

⁰

has been replaced by G

⁰₀

. This relationship does not have a physical basis, but is rather made up to meet the requirements of rate-independency of soils and other materials. The rate-independent Kelvin model is also used in structural damping models. Similarly as in eq. 2.4, we get:

G = (1 + iη)G ¯ (2.58)

or, expressed in Young’s modulus and damping ratio:

E = (1 + i2ξ)E ¯ (2.59)

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